The linearization problem described above is one of the most fundamental ones in the theory of dynamical systems and has a long history (of which we do not attempt to give a complete survey). In the context of autonomous dynamics, the problem of formulating sufficient conditions under which the conjugacy exhibits higher regularity properties was first considered in the pioneering works of Sternberg [1]}, [2]} who proved in the 1950s that
for each \(r\in \mathbb {N}\)  there is an integer \(k\in \mathbb {N}\)  such that \(C^k\)  hyperbolic diffeomorphisms with non-resonance conditions up to order \(k\)  admit a \(C^r\)  linearization.
We also mention the fundamental works of Hartman [3]} and Grobman [4]} who proved, independently, in the 1960s that \(C^1\)  hyperbolic diffeomorphisms can be \(C^0\)  linearized near the hyperbolic ﬁxed point. It shows that if one is only interested in the case where \(r=0\)  in the Sternberg's theorem, then one can take \(k=1\)  and, moreover, no non-resonance condition is needed (see Section  for more details). We note that infinite dimensional versions of this result were obtained independently by Palis ([5]}) and Pugh ([6]}).
